**Differential Equations With Mathematica, Fifth Edition**

Martha L. Abell and James P. Braselton

**Contents**

Preface ix

Introduction to differential equations

1.1 Definitions and concepts 1

1.2 Solutions of differential equations 4

1.3 Initial- and boundary-value problems 12

1.4 Direction fields 18

1.4.1 Creating interactive applications 27

2. First-order ordinary differential equations

2.1 Theory of first-order equations: a brief discussion 29

2.2 Separation of variables 33

2.3 Homogeneous equations 42

2.4 Exact equations 46

2.5 **Linear equations** 50

2.5.1 Integrating factor approach 51

2.5.2 Variation of parameters and the method of undetermined coefficients 56

2.6 Numerical approximations

of solutions to first-order

equations 60

2.6.1 Built-in methods 60

2.6.2 Other numerical

methods 63

Application: modeling

the spread of a disease 77

Applications of

first-order equations

3.1 Orthogonal trajectories 83

3.2 Population growth and

decay 90

3.2.1 The Malthus model 90

3.2.2 The logistic equation 95

3.3 Newton’s law of cooling 104

3.4 Free-falling bodies 108

Higher-order linear

differential equations

4.1 Preliminary definitions and

notation 115

4.1.1 Introduction 115

4.1.2 The nth-order

ordinary linear

differential equation 119

4.1.3 Fundamental set of

solutions 124

4.1.4 Existence of a

fundamental set of

solutions 127

4.1.5 Reduction of order 128

4.2 Solving homogeneous

equations with constant

coefficients 131

4.2.1 Second-order

equations 131

4.2.2 Higher-order

equations 135

4.3 Introduction to solving

nonhomogeneous equations 141

4.4 Nonhomogeneous

equations with constant

coefficients: the method of

undetermined coefficients 145

4.4.1 Second-order

equations 147

4.4.2 Higher-order

equations 158

4.5 **Nonhomogeneous equations** with constant coefficients: variation of parameters 164

4.5.1 Second-order

equations 164

4.5.2 Higher-order

nonhomogeneous

equations 167

4.6 Cauchy–Euler equations 170

4.6.1 Second-order

Cauchy–Euler

equations 170

4.6.2 Higher-order

Cauchy–Euler

equations 173

4.6.3 Variation of

parameters 177

4.7 Series solutions 179

4.7.1 Power series solutions

about ordinary points 179

4.7.2 Series solutions about

regular singular points 189

4.7.3 Method of Frobenius 190

Application: zeros of

the Bessel functions of

the first kind 200

4.8 Nonlinear equations 205

Applications of

higher-order differential

equations

5.1 Harmonic motion 221

5.1.1 Simple harmonic motion 221

5.1.2 Damped motion 228

5.1.3 Forced motion 238

5.1.4 Soft springs 250

5.1.5 Hard springs 253

5.1.6 Aging springs 254

Application: hearing

beats and resonance 255

5.2 The pendulum problem 256

5.3 Other applications 265

5.3.1 L-R-C circuits 265

5.3.2 Deflection of a beam 268

5.3.3 Bodé plots 270

5.3.4 The catenary 273

Systems of ordinary

differential equations

6.1 Review of matrix algebra

and calculus 283

6.1.1 Defining nested lists,

matrices, and vectors 283

6.1.2 Extracting elements of

matrices 287

6.1.3 Basic computations

with matrices 288

6.1.4 Systems of linear

equations 290

6.1.5 Eigenvalues and

eigenvectors 292

6.1.6 Matrix calculus 296

6.2 Systems of equations:

preliminary definitions and

theory 297

6.2.1 Preliminary theory 301

6.2.2 Linear systems 308

6.3 Homogeneous linear

systems with constant

coefficients 315

6.3.1 Distinct real

eigenvalues 315

6.3.2 Complex conjugate

eigenvalues 320

6.3.3 Solving initial-value

problems 325

6.3.4 Repeated eigenvalues 327

6.4 Nonhomogeneous

first-order systems:

undetermined coefficients,

variation of parameters, and

the matrix exponential 333

6.4.1 Undetermined

coefficients 334

6.4.2 Variation of

parameters 337

6.4.3 The matrix

exponential 342

6.5 Numerical methods 348

6.5.1 Built-in methods 349

6.5.2 Euler’s method 356

6.5.3 Runge–Kutta method 360

6.6 Nonlinear systems,

linearization, and

classification of equilibrium

points 362

6.6.1 Real distinct

eigenvalues 363

6.6.2 Repeated eigenvalues 368

6.6.3 Complex conjugate

eigenvalues 371

6.6.4 Nonlinear systems 374

Applications of systems

of ordinary differential

equations

7.1 Mechanical and electrical

problems with first-order

linear systems 385

7.1.1 L-R-C circuits with

loops 385

7.1.2 L-R-C circuit with

one loop 385

7.1.3 L-R-C circuit with

two loops 387

7.1.4 Spring–mass systems 390

7.2 Diffusion and population

problems with first-order

linear systems 391

7.2.1 Diffusion through a

membrane 391

7.2.2 Diffusion through a

double-walled

membrane 393

7.2.3 Population problems 396

7.3 Applications that lead to

nonlinear systems 399

7.3.1 Biological systems:

predator–prey

interactions, the

Lotka–Volterra system,

and food chains in the

chemostat 400

7.3.2 Physical systems:

variable damping 414

7.3.3 Differential geometry:

curvature 418

Laplace transform

methods

8.1 The Laplace transform 423

8.1.1 Definition of the

Laplace transform 423

8.1.2 Exponential order,

jump discontinuities,

and piecewise

continuous functions 426

8.1.3 Properties of the

Laplace transform 428

8.2 The inverse Laplace

transform 432

8.2.1 Definition of the

inverse Laplace

transform 432

8.2.2 Laplace transform of

an integral 438

8.3 Solving initial-value

problems with the Laplace

transform 439

8.4 Laplace transforms of step

and periodic functions 444

8.4.1 Piecewise defined

functions: the unit

step function 444

8.4.2 Solving initial-value

problems with

piecewise continuous

forcing functions 447

8.4.3 Periodic functions 449

8.4.4 Impulse functions: the

delta function 457

8.5 The convolution theorem 462

8.5.1 The convolution

theorem 462

8.5.2 Integral and

integrodifferential

equations 464

8.6 Applications of Laplace

transforms, Part I 466

8.6.1 Spring–mass systems

revisited 466

8.6.2 L-R-C circuits

revisited 469

8.6.3 Population problems

revisited 474

Application: the

tautochrone 476

8.7 Laplace transform methods

for systems 478

8.8 Applications of Laplace

transforms, Part II 488

8.8.1 Coupled spring–mass

systems 488

8.8.2 The double pendulum 493

Application: free

vibration of a

three-story building 496

Eigenvalue problems and

Fourier series

9.1 Boundary-value problems,

eigenvalue problems, and

Sturm–Liouville problems 503

9.1.1 Boundary-value

problems 503

9.1.2 Eigenvalue problems 505

9.1.3 Sturm–Liouville

problems 508

9.2 Fourier sine series and

cosine series 510

9.2.1 Fourier sine series 510

9.2.2 Fourier cosine series 515

9.3 Fourier series 518

9.3.1 Fourier series 518

9.3.2 Even, odd, and

periodic extensions 525

9.3.3 Differentiation and

integration of Fourier

series 530

9.3.4 Parseval’s equality 534

9.4 Generalized Fourier series 535

10.Partial differential

equations

10.1 Introduction to partial

differential equations and

separation of variables 545

10.1.1 Introduction 545

10.1.2 Separation of

variables 546

10.2 The one-dimensional heat

equation 548

10.2.1 The heat equation

with homogeneous

boundary conditions 548

10.2.2 Nonhomogeneous

boundary conditions 551

10.2.3 Insulated boundary 554

10.3 The one-dimensional wave

equation 557

10.3.1 The wave equation 557

10.3.2 D’Alembert’s solution 562

10.4 Problems in two dimensions:

Laplace’s equation 565

10.4.1 Laplace’s equation 565

10.5 Two-dimensional problems

in a circular region 570

10.5.1 Laplace’s equation in

a circular region 570

10.5.2 The wave equation in

a circular region 573

Bibliography 585

Index 587